Nearest-neighbor walks with low predictability profile and percolation in $2+\epsilon$ dimensions

Abstract

A few years ago, Grimmett, Kesten and Zhang proved that for supercritical bond percolation on $\mathbf{Z}^3$, simple random walk on the infinite cluster is a.s. transient. We generalize this result to a class of wedges in $\mathbf{Z}^3$ including, for any $\varepsilon \epsilon (0, 1)$, the wedge $\mathscr{W}_{\varepsilon} = {(x, y, z) \epsilon \mathbf{Z}^3: x \geq 0, |z| \leq x^{\varepsilon}}$ which can be thought of as representing a $(2 + \varepsilon)$-dimensional lattice. Our proof builds on recent work of Benjamini, Pemantle and Peres, and involves the construction of finite-energy flows using nearest-neighbor walks on Z with low predictability profile. Along the way, we obtain some new results on attainable decay rates for predictability profiles of nearest-neighbor walks.